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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.stat_tut.dist_params"></a><a class="link" href="dist_params.html" title="Discrete Probability Distributions">Discrete Probability
      Distributions</a>
</h3></div></div></div>
<p>
        Note that the <a href="http://en.wikipedia.org/wiki/Discrete_probability_distribution" target="_top">discrete
        distributions</a>, including the binomial, negative binomial, Poisson
        &amp; Bernoulli, are all mathematically defined as discrete functions: only
        integral values of the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
        variate</a> are envisaged and the functions are only defined at these
        integral values. However because the method of calculation often uses continuous
        functions, it is convenient to treat them as if they were continuous functions,
        and permit non-integral values of their parameters.
      </p>
<p>
        To enforce a strict mathematical model, users may use floor or ceil functions
        on the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a>,
        prior to calling the distribution function, to enforce integral values.
      </p>
<p>
        For similar reasons, in continuous distributions, parameters like degrees
        of freedom that might appear to be integral, are treated as real values (and
        are promoted from integer to floating-point if necessary). In this case however,
        that there are a small number of situations where non-integral degrees of
        freedom do have a genuine meaning.
      </p>
<p>
        Generally speaking there is no loss of performance from allowing real-values
        parameters: the underlying special functions contain optimizations for integer-valued
        arguments when applicable.
      </p>
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<th align="left">Caution</th>
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<p>
          The quantile function of a discrete distribution will by default return
          an integer result that has been <span class="emphasis"><em>rounded outwards</em></span>.
          That is to say lower quantiles (where the probability is less than 0.5)
          are rounded downward, and upper quantiles (where the probability is greater
          than 0.5) are rounded upwards. This behaviour ensures that if an X% quantile
          is requested, then <span class="emphasis"><em>at least</em></span> the requested coverage
          will be present in the central region, and <span class="emphasis"><em>no more than</em></span>
          the requested coverage will be present in the tails.
        </p>
<p>
          This behaviour can be changed so that the quantile functions are rounded
          differently, or even return a real-valued result using <a class="link" href="../pol_overview.html" title="Policy Overview">Policies</a>.
          It is strongly recommended that you read the tutorial <a class="link" href="../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
          Quantiles of Discrete Distributions</a> before using the quantile function
          on a discrete distribution. The <a class="link" href="../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
          docs</a> describe how to change the rounding policy for these distributions.
        </p>
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      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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